Augmented pseudo-marginal Metropolis-Hastings for partially observed diffusion processes

We consider the problem of inference for nonlinear, multivariate diffusion processes, satisfying Itô stochastic differential equations (SDEs), using data at discrete times that may be incomplete and subject to measurement error. Our starting point is a state-of-the-art correlated pseudo-marginal Metropolis–Hastings algorithm, that uses correlated particle filters to induce strong and positive correlation between successive likelihood estimates. However, unless the measurement error or the dimension of the SDE is small, correlation can be eroded by the resampling steps in the particle filter. We therefore propose a novel augmentation scheme, that allows for conditioning on values of the latent process at the observation times, completely avoiding the need for resampling steps. We integrate over the uncertainty at the observation times with an additional Gibbs step. Connections between the resulting pseudo-marginal scheme and existing inference schemes for diffusion processes are made, giving a unified inference framework that encompasses Gibbs sampling and pseudo marginal schemes. The methodology is applied in three examples of increasing complexity. We find that our approach offers substantial increases in overall efficiency, compared to competing methods

Efficiency of delayed-acceptance random walk Metropolis algorithms

Delayed-acceptance Metropolis-Hastings (DA-MH) and delayed-acceptance pseudo-marginal Metropolis-Hastings (DAPsMMH) algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased stochastic approximation thereof, but a computationally cheap deterministic approximation is available. An initial accept-reject stage uses the cheap approximation for computing the Metropolis-Hastings ratio; proposals which are accepted at this stage are then subjected to a further accept-reject step which corrects for the error in the approximation. Since the expensive posterior, or the approximation thereof, is only evaluated for proposals which are accepted at the first stage, the cost of the algorithm is reduced. We focus on the random walk Metropolis (RWM) and consider the DAPsMRWM, of which the DARWM is a special case. We provide a framework for incorporating relatively general deterministic approximations into the theoretical analysis of high-dimensional targets. Then, justified by a limiting diffusion argument, we develop theoretical expressions for limiting efficiency and acceptance rates in high dimension. The results provide insight into the effect of the accuracy of the deterministic approximation, the scale of the RWM jump and the nature of the stochastic approximation on the efficiency of the delayed acceptance algorithm. The predicted properties are verified against simulation studies, all of which are strictly outside of the domain of validity of our limit results. The theory also informs a practical strategy for algorithm tuning

Accelerating inference for stochastic kinetic models

Stochastic kinetic models (SKMs) are increasingly used to account for the inherent stochasticity exhibited by interacting populations of species in areas such as epidemiology, population ecology and systems biology. Species numbers are modelled using a continuous-time stochastic process, and, depending on the application area of interest, this will typically take the form of a Markov jump process or an It\^o diffusion process. Widespread use of these models is typically precluded by their computational complexity. In particular, performing exact fully Bayesian inference in either modelling framework is challenging due to the intractability of the observed data likelihood, necessitating the use of computationally intensive techniques such as particle Markov chain Monte Carlo (particle MCMC). It is proposed to increase the computational and statistical efficiency of this approach by leveraging the tractability of an inexpensive surrogate derived directly from either the jump or diffusion process. The surrogate is used in three ways: in the design of a gradient-based parameter proposal, to construct an appropriate bridge and in the first stage of a delayed-acceptance step. The resulting approach, which exactly targets the posterior of interest, offers substantial gains in efficiency over a standard particle MCMC implementation.Comment: 29 page

Using extreme value theory to evaluate the leading pedestrian interval road safety intervention

Improving road safety is hugely important with the number of deaths on the world's roads remaining unacceptably high; an estimated 1.35 million people die each year as a result of road traffic collisions (WHO, 2020). Current practice for treating collision hotspots is almost always reactive: once a threshold level of collisions has been overtopped during some pre-determined observation period, treatment is applied (e.g. road safety cameras). Traffic collisions are rare, so prolonged observation periods are necessary. However, traffic conflicts are more frequent and are a margin of the social cost; hence, traffic conflict before/after studies can be conducted over shorter time periods. We investigate the effect of implementing the leading pedestrian interval (LPI) treatment (Van Houten et al. 2000) at signalised intersections as a safety intervention in a city in north America. Pedestrian-vehicle traffic conflict data were collected from treatment and control sites during the before and after periods. We implement a before/after study on post-encroachment times (PETs) where small PET values denote a near-miss. Hence, extreme value theory is employed to model extremes of our PET processes, with adjustments to the usual modelling framework to account for temporal dependence and treatment effects.Comment: 16 page

Using extreme value theory to evaluate the leading pedestrian interval road safety intervention

Improving road safety is hugely important with the number of deaths on the world's roads remaining unacceptably high; an estimated 1.3 million people die each year as a result of road traffic collisions. Current practice for treating collision hotspots is almost always reactive: once a threshold level of collisions has been overtopped during some pre‐determined observation period, treatment is applied (e.g., road safety cameras). Traffic collisions are rare, so prolonged observation periods are necessary. However, traffic conflicts are more frequent and are a margin of the social cost; hence, traffic conflict before/after studies can be conducted over shorter time periods. We investigate the effect of implementing the leading pedestrian interval treatment at signalised intersections as a safety intervention in a city in north America. Pedestrian‐vehicle traffic conflict data were collected from treatment and control sites during the before and after periods. We implement a before/after study on post‐encroachment times (PETs) where small PET values denote ‘near‐misses’. Hence, extreme value theory is employed to model extremes of our PET processes, with adjustments to the usual modelling framework to account for temporal dependence and treatment effects

Bayesian inference for a spatio-temporal model of road traffic collision data

Improving road safety is hugely important with the number of deaths on the world’s roads remaining unacceptably high; an estimated 1.35 million people die each year (WHO, 2020). Current practice for treating collision hotspots is almost always reactive: once a threshold level of collisions has been exceeded during some predetermined observation period, treatment is applied (e.g. road safety cameras). However, more recently, methodology has been developed to predict collision counts at potential hotspots in future time periods, with a view to a more proactive treatment of road safety hotspots. Dynamic linear models provide a flexible framework for predicting collisions and thus enabling such a proactive treatment. In this paper, we demonstrate how such models can be used to capture both seasonal variability and spatial dependence in time dependent collision rates at several locations. The model allows for within- and out-of-sample forecasting for locations which are fully observed and for locations where some data are missing. We illustrate our approach using collision rate data from 8 Traffic Administration Zones in the US, and find that the model provides a good description of the underlying process and reasonable forecast accuracy

Ensemble MCMC: Accelerating Pseudo-Marginal MCMC for State Space Models using the Ensemble Kalman Filter

Particle Markov chain Monte Carlo (pMCMC) is now a popular method for performing Bayesian statistical inference on challenging state space models (SSMs) with unknown static parameters. It uses a particle filter (PF) at each iteration of an MCMC algorithm to unbiasedly estimate the likelihood for a given static parameter value. However, pMCMC can be computationally intensive when a large number of particles in the PF is required, such as when the data are highly informative, the model is misspecified and/or the time series is long. In this paper we exploit the ensemble Kalman filter (EnKF) developed in the data assimilation literature to speed up pMCMC. We replace the unbiased PF likelihood with the biased EnKF likelihood estimate within MCMC to sample over the space of the static parameter. On a wide class of different non-linear SSM models, we demonstrate that our extended ensemble MCMC (eMCMC) methods can significantly reduce the computational cost whilst maintaining reasonable accuracy. We also propose several extensions of the vanilla eMCMC algorithm to further improve computational efficiency. Computer code to implement our methods on all the examples can be downloaded from https://github.com/cdrovandi/Ensemble-MCMC

Accelerating Bayesian inference for stochastic epidemic models using incidence data

We consider the case of performing Bayesian inference for stochastic epidemic compartment models, using incomplete time course data consisting of incidence counts that are either the number of new infections or removals in time intervals of fixed length. We eschew the most natural Markov jump process representation for reasons of computational efficiency, and focus on a stochastic differential equation representation. This is further approximated to give a tractable Gaussian process, that is, the linear noise approximation (LNA). Unless the observation model linking the LNA to data is both linear and Gaussian, the observed data likelihood remains intractable. It is in this setting that we consider two approaches for marginalising over the latent process: a correlated pseudo-marginal method and analytic marginalisation via a Gaussian approximation of the observation model. We compare and contrast these approaches using synthetic data before applying the best performing method to real data consisting of removal incidence of oak processionary moth nests in Richmond Park, London. Our approach further allows comparison between various competing compartment models